Some Equalities and Inequalities for the Hermitian Moore-Penrose Inverse of Triple Matrix Product with Applications

DOI：10.3770/j.issn:2095-2651.2015.03.010

 作者 单位 田永革 中央财经大学中国经济与管理研究院, 北京 100081 郭文星 中央财经大学数学与统计学院, 北京 100081

通过矩阵的秩和惯量公式研究矩阵广义逆之间的关系,建立了三个矩阵乘积的厄米特Moore-Penrose逆的一些等式与不等式.作为应用,给出了若干两个矩阵之和厄米特Moore-Penrose逆的等式与不等式.

We investigate relationships between the Moore-Penrose inverse \$(ABA^{*})^{\dag}\$ and the product \$[(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}\$ through some rank and inertia formulas for the difference of \$(ABA^{*})^{\dag} - [(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}\$, where \$B\$ is Hermitian matrix and \$(AB)^{(1,2,3)}\$ is a \$\{1,\, 2,\,3\}\$-inverse of \$AB\$. We show that there always exists an \$(AB)^{(1,2,3)}\$ such that \$(ABA^{*})^{\dag}=[(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}\$ holds. In addition, we also establish necessary and sufficient conditions for the two inequalities \$(ABA^{*})^{\dag} \succ [(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}\$ and \$(ABA^{*})^{\dag} \prec [(AB)^{(1,2,3)}]^{*}B(AB)^{(1,2,3)}\$ to hold in the L\"owner partial ordering. Some variations of the equalities and inequalities are also presented. In particular, some equalities and inequalities for the Moore-Penrose inverse of the sum \$A + B\$ of two Hermitian matrices \$A\$ and \$B\$ are established.